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Lesson Explainer: Recursive Formula of a Sequence Mathematics • 9th Grade

In this explainer, we will learn how to find the recursive formula of a sequence.

Recall that a sequence is just a list of numbers. Some common types of sequence that we meet at this level include arithmetic sequences, where the difference between consecutive terms is constant, and geometric sequences, where there is a common ratio between consecutive terms. Here, we will discuss a powerful method for representing a variety of different sequences that involves knowing the rule to get from one term to the next in a sequence. This means we can then describe the sequence in terms of a recursive formula.

Definition: Recursive Formula of a Sequence

A recursive formula (sometimes called a recurrence relation) is a formula that defines each term of a sequence using a preceding term or terms.

A recursive formula of the form 𝑇=𝑓(𝑇) defines each term of a sequence as a function of the previous term. To generate a sequence from its recursive formula, we need to know the first term in the sequence, π‘‡οŠ§.

If we know the first term, π‘‡οŠ§, and the recursive formula 𝑇=𝑓(𝑇), we can use the formula with 𝑛=1 to derive the value of π‘‡οŠ¨ from π‘‡οŠ§. Once we know the value of π‘‡οŠ¨, we can use the formula with 𝑛=2 to derive the value of π‘‡οŠ© from π‘‡οŠ¨, and so on. In this way, we can build up the sequence until it has as many terms as we wish.

This process is best illustrated through some specific examples. Suppose we are asked to find the first four terms of the sequence defined by the recursive formula 𝑇=𝑇+1,𝑇=10.

We already know the first term, which is 𝑇=10.

To find the second term, π‘‡οŠ¨, we substitute 𝑛=1 into the recursive formula 𝑇=𝑇+1 and use the fact that 𝑇=10 to get 𝑇=𝑇+1=10+1=11.

Similarly, to find π‘‡οŠ©, we substitute 𝑛=2 into the formula and use the fact that 𝑇=11 to get 𝑇=𝑇+1=11+1=12.

Finally, to find 𝑇οŠͺ, we substitute 𝑛=3 into the formula and use the fact that 𝑇=12 to get 𝑇=𝑇+1=12+1=13.οŠͺ

Therefore, the first four terms of this sequence are (10,11,12,13). Note that this is actually an arithmetic sequence with a first term of 10 and a common difference of 1.

As another example, suppose we are asked to find the first five terms of the sequence defined by the recursive formula 𝑇=π‘‡βˆ’4,𝑇=8.

Clearly, the first term is 𝑇=8.

To find the second term, π‘‡οŠ¨, we substitute 𝑛=1 into the recursive formula 𝑇=π‘‡βˆ’4 and use the fact that 𝑇=8 to get 𝑇=π‘‡βˆ’4=8βˆ’4=4.

Similarly, to find π‘‡οŠ©, we substitute 𝑛=2 into the formula and use the fact that 𝑇=4 to get 𝑇=π‘‡βˆ’4=4βˆ’4=0.

To find 𝑇οŠͺ, we substitute 𝑛=3 into the formula and use the fact that 𝑇=0 to get 𝑇=π‘‡βˆ’4=0βˆ’4=βˆ’4.οŠͺ

Finally, to find π‘‡οŠ«, we substitute 𝑛=4 into the formula and use the fact that 𝑇=βˆ’4οŠͺ to get 𝑇=π‘‡βˆ’4=βˆ’4βˆ’4=βˆ’8.οŠͺ

Therefore, the first five terms of this sequence are (8,4,0,βˆ’4,βˆ’8). Note that this is actually an arithmetic sequence with a first term of 8 and a common difference of βˆ’4. The strength of recursive formulas is that they enable us to describe many different types of sequence, including arithmetic sequences, geometric sequences, and others that we will meet later on in this explainer.

Let us now try an example to practice this skill.

Example 1: Finding the First Five Terms of a Sequence Using Its Recursive Formula

Find the first five terms of the sequence with general term 𝑇=𝑇+5, where 𝑛β‰₯1 and 𝑇=βˆ’13.

Answer

Recall that a recursive formula of the form 𝑇=𝑓(𝑇) defines each term of a sequence as a function of the previous term. To generate a sequence from its recursive formula, we need to know the first term in the sequence, that is, π‘‡οŠ§.

Here, we are given the first term 𝑇=βˆ’13 together with the recursive formula 𝑇=𝑇+5.

To find the second term, π‘‡οŠ¨, we substitute 𝑛=1 into the recursive formula and use the fact that 𝑇=βˆ’13 to get 𝑇=𝑇+5=βˆ’13+5=βˆ’8.

Similarly, by substituting 𝑛=2, 3, and 4 into the given recursive formula, we get 𝑇=𝑇+5=βˆ’8+5=βˆ’3,𝑇=𝑇+5=βˆ’3+5=2,𝑇=𝑇+5=2+5=7.οŠͺοŠͺ

We deduce that the first five terms of this sequence are (βˆ’13,βˆ’8,βˆ’3,2,7). Note that this is actually an arithmetic sequence with a first term of βˆ’13 and a common difference of 5.

In some questions, we are given the first few terms of a recursive sequence and must work backward from those terms to find a recursive formula. For example, suppose we are asked to find a recursive formula for this sequence: 3,1,βˆ’1,βˆ’3,βˆ’5,….

We start by inspecting the sequence to see if we can spot the pattern of relationships between the terms. A sensible approach is to check first if each term can be obtained from the previous one by adding or subtracting a constant.

In this case, the terms are decreasing steadily, so we have to subtract. To get from the first term 𝑇=3 to the second term 𝑇=1, we subtract 2. Similarly, to get from the second term 𝑇=1 to the third term 𝑇=βˆ’1, we subtract 2, and so on.

Therefore, a recursive formula for this sequence is 𝑇=π‘‡βˆ’2. To enable someone else to generate this sequence, we also need to state that 𝑇=3.

Now, let us try another example. Suppose we are asked to find a recursive formula for this sequence: 2,10,50,250,1250,….

Again, we start by inspecting the sequence to see if we can spot the pattern of relationships between the terms. In this case, as the terms are increasing, we first check if we can get from one term to the next by adding a constant. To get from the first term 𝑇=2 to the second term 𝑇=10, we add 8. However, to get from the second term 𝑇=10 to the third term 𝑇=50, we add 40, to get from the third term 𝑇=50 to the fourth term 𝑇=250οŠͺ, we add 200, and to get from the fourth term 𝑇=250οŠͺ to the fifth term 𝑇=1250, we add 1β€Žβ€‰β€Ž000.

This clearly shows that any recursive formula for the sequence cannot involve adding a constant, because the terms increase by ever larger amounts. Such a pattern suggests we should instead check if we can get from one term to the next by multiplying by a constant. To get from the first term 𝑇=2 to the second term 𝑇=10, we multiply by 5. Similarly, to get from the second term 𝑇=10 to the third term 𝑇=50, we multiply by 5, and so on.

Therefore, a recursive formula for this sequence is 𝑇=5π‘‡οŠοŠ°οŠ§οŠ. To enable someone else to generate this sequence, we also need to state that 𝑇=2.

Let us now try to apply this methodology in the next example.

Example 2: Finding the Recursive Formula of a Sequence given the First Seven Terms

Find a recursive formula for the sequence 486,162,54,18,6,2,23.

Answer

Recall that a recursive formula of the form 𝑇=𝑓(𝑇) defines each term of a sequence as a function of the previous term. To generate a sequence from its recursive formula, we need to know the first term in the sequence, that is, π‘‡οŠ§.

We start by inspecting the sequence to see if we can spot the pattern of relationships between the terms. First, we check if each term can be obtained from the previous one by adding or subtracting a constant. Here, the terms are decreasing, so to get from one term to the next, we need to subtract.

However, the above diagram clearly shows that any recursive formula for the sequence cannot involve subtracting a constant, because the terms decrease by ever smaller amounts. Such a pattern suggests that we should instead check if we can get from one term to the next by multiplying by a constant.

To find the value of this constant, we can use the following idea.

In general, to get from the first term π‘‡οŠ§ to the second term π‘‡οŠ¨, we must multiply by π‘‡π‘‡οŠ¨οŠ§, because 𝑇×𝑇𝑇=π‘‡οŠ§οŠ¨οŠ§οŠ¨.

Similarly, to get from the second term π‘‡οŠ¨ to the third term π‘‡οŠ©, we must multiply by π‘‡π‘‡οŠ©οŠ¨, because 𝑇×𝑇𝑇=π‘‡οŠ¨οŠ©οŠ¨οŠ©.

Continuing in this way, we can check the multiplier values for each pair of consecutive terms in the sequence and see if they agree. Notice that as the sequence is decreasing, we will expect any multiplier, which is a number of the form π‘‡π‘‡οŠοŠ°οŠ§οŠ with 𝑇<π‘‡οŠοŠ°οŠ§οŠ, to be greater than 0 but less than 1.

To get from the first term 𝑇=486 to the second term 𝑇=162, we know that 𝑇×𝑇𝑇=π‘‡οŠ§οŠ¨οŠ§οŠ¨. This means we must multiply π‘‡οŠ§ by 𝑇𝑇=162486=13.

To get from the second term 𝑇=162 to the third term 𝑇=54, we must multiply by 𝑇𝑇=54162=13.

Further calculations for the remaining pairs of consecutive terms show that we obtain the same value in every case, so we have the following arrangement.

Note that multiplying each term by the fraction 13 is the same as dividing it by 3. Even so, it is usually more straightforward to think of this as a process of repeated multiplication rather than repeated division.

Hence, a recursive formula for this sequence is 𝑇=13𝑇,𝑇=486. Note that this is a geometric sequence with a first term of 486 and a common ratio of 13.

Sometimes, we meet sequences that are defined by using an 𝑛th term rule. For instance, the sequence defined by the 𝑛th term rule 𝑇=2π‘›οŠ for 𝑛β‰₯1 is just 2,4,6,8,10,…. We may be asked to rewrite a sequence given in this form by using a recursive formula. Here is an example of this type.

Example 3: Writing the Recursive Formula of a Sequence given Its 𝑛th Term Formula

By writing down the first four terms, or otherwise, find the recursive formula that defines the sequence 𝑇=𝑛+14 for 𝑛β‰₯1.

Answer

Recall that a recursive formula of the form 𝑇=𝑓(𝑇) defines each term of a sequence as a function of the previous term. To generate a sequence from its recursive formula, we need to know the first term in the sequence, that is, π‘‡οŠ§.

We start by substituting 𝑛=1 into the 𝑛th term rule 𝑇=𝑛+14 to get the first term of the sequence. This gives 𝑇=1+14=24=12.

Similarly, by substituting 𝑛=2, 3, and 4 into the 𝑛th term rule, we get the second, third, and fourth terms as follows: 𝑇=2+14=34,𝑇=3+14=44=1,𝑇=4+14=54.οŠͺ

By listing the first four terms as fractions with a common denominator of 4, we can see an obvious pattern emerging for the sequence, as shown below.

To get from one term to the next, we must add 14. Therefore, we can immediately write down the following recursive formula for the sequence, remembering to include the first term: 𝑇=𝑇+14,𝑇=12.

Another way to characterize sequences is by classifying them as increasing, decreasing, or periodic. We will need the following definitions.

Definition: Increasing, Decreasing, and Periodic Sequences

A sequence is increasing if 𝑇>π‘‡οŠοŠ°οŠ§οŠ for all natural numbers 𝑛.

A sequence is decreasing if 𝑇<π‘‡οŠοŠ°οŠ§οŠ for all natural numbers 𝑛.

A sequence is periodic if its terms repeat in a cycle. For any periodic sequence, there is an integer π‘˜ such that 𝑇=π‘‡οŠοŠ°ο‡οŠ for all natural numbers 𝑛. The integer π‘˜ is called the order (or period) of the sequence.

It is important to realize that not all sequences can be classified in this way. For instance, the sequence 1,βˆ’2,4,βˆ’8,16,βˆ’32,… can be defined by the recursive formula 𝑇=βˆ’2π‘‡οŠοŠ°οŠ§οŠ with 𝑇=1, but it is neither increasing, decreasing, nor periodic. The terms flip between positive and negative, but since the terms continually increase in magnitude, there is no constant π‘˜ such that 𝑇=π‘‡οŠοŠ°ο‡οŠ. However, for the remainder of this explainer, we will focus on sequences that do fit into one of these three categories.

In our next example, we must use the first term and recursive formula of a sequence to work out whether the sequence is increasing, decreasing, or periodic.

Example 4: Identifying Whether a Sequence Is Increasing, Decreasing, or Periodic Using Its Recursive Formula

A sequence is given by the recursive formula 𝑇=𝑇+6,𝑇=βˆ’3. Is this sequence increasing, decreasing, or periodic? If it is periodic, state its order.

Answer

Recall that a recursive formula of the form 𝑇=𝑓(𝑇) defines each term of a sequence as a function of the previous term. To generate a sequence from its recursive formula, we need to know the first term in the sequence, that is, π‘‡οŠ§.

Here, we are given the first term 𝑇=βˆ’3 together with the recursive formula 𝑇=𝑇+6.

To find the second term, π‘‡οŠ¨, we substitute 𝑛=1 into the formula and use the fact that 𝑇=βˆ’3 to get 𝑇=𝑇+6=βˆ’3+6=3.

Then, to find the third term, π‘‡οŠ©, we substitute 𝑛=2 into the formula and use the fact that 𝑇=3 to get 𝑇=𝑇+6=3+6=9.

Similarly, by substituting 𝑛=3, 4, and 5 into the formula, we get 𝑇=𝑇+6=9+6=15,𝑇=𝑇+6=15+6=21,𝑇=𝑇+6=21+6=27.οŠͺοŠͺ

Recall that a sequence is increasing if 𝑇>π‘‡οŠοŠ°οŠ§οŠ for all natural numbers 𝑛. A sequence is decreasing if 𝑇<π‘‡οŠοŠ°οŠ§οŠ for all natural numbers 𝑛. Finally, a sequence is periodic if its terms repeat in a cycle.

Clearly, the sequence βˆ’3,3,9,15,21,27,… satisfies 𝑇>π‘‡οŠοŠ°οŠ§οŠ for all natural numbers 𝑛, so we conclude that it is increasing.

In fact, we could have immediately reached the same conclusion simply by inspecting the recursive formula of the sequence. As 𝑇=𝑇+6, the value of each term is 6 more than the previous one. Thus, 𝑇>π‘‡οŠοŠ°οŠ§οŠ, so the sequence must be increasing.

In general, for any positive number π‘š, sequences with recursive formulas of the form 𝑇=𝑇+π‘šοŠοŠ°οŠ§οŠ are increasing, while those with recursive formulas of the form 𝑇=π‘‡βˆ’π‘šοŠοŠ°οŠ§οŠ are decreasing. It is often more complicated to spot periodic sequences from their recursive formulas.

Our final example tests the same process of recognition.

Example 5: Identifying Whether a Sequence Is Increasing, Decreasing, or Periodic Using Its Recursive Formula

A sequence is given by the recursive formula 𝑇=βˆ’π‘‡οŠοŠ°οŠ§οŠ, 𝑇=8. Is this sequence increasing, decreasing, or periodic? If it is periodic, state its order.

Answer

Recall that a recursive formula of the form 𝑇=𝑓(𝑇) defines each term of a sequence as a function of the previous term. To generate a sequence from its recursive formula, we need to know the first term in the sequence, that is, π‘‡οŠ§.

Here, we are given the first term 𝑇=8 together with the recursive formula 𝑇=βˆ’π‘‡οŠοŠ°οŠ§οŠ.

To find the second term, π‘‡οŠ¨, we substitute 𝑛=1 into the formula and use the fact that 𝑇=8 to get 𝑇=βˆ’π‘‡=βˆ’8.

Then, to find the third term, π‘‡οŠ©, we substitute 𝑛=2 into the formula and use the fact that 𝑇=βˆ’8 to get 𝑇=βˆ’π‘‡=βˆ’(βˆ’8)=8.

Similarly, by substituting 𝑛=3, 4, and 5 into the formula, we get 𝑇=βˆ’π‘‡=βˆ’8,𝑇=βˆ’π‘‡=βˆ’(βˆ’8)=8,𝑇=βˆ’π‘‡=βˆ’8.οŠͺοŠͺ

Recall that a sequence is increasing if 𝑇>π‘‡οŠοŠ°οŠ§οŠ for all natural numbers 𝑛. A sequence is decreasing if 𝑇<π‘‡οŠοŠ°οŠ§οŠ for all natural numbers 𝑛. Finally, a sequence is periodic if its terms repeat in a cycle.

Clearly, the terms of the sequence 8,βˆ’8,8,βˆ’8,8,βˆ’8,… repeat in a cycle, so the sequence is periodic.

For any periodic sequence, there is an integer π‘˜ such that 𝑇=π‘‡οŠοŠ°ο‡οŠ for all natural numbers 𝑛. The integer π‘˜ is called the order (or period) of the sequence. In this case, we have 𝑇=𝑇=𝑇=8 and 𝑇=𝑇=𝑇=βˆ’8οŠͺ. So, the sequence splits into 2 separate sets of terms, with the first set satisfying 𝑇=π‘‡οŠοŠ°οŠ¨οŠ for odd 𝑛 and the second set satisfying 𝑇=π‘‡οŠοŠ°οŠ¨οŠ for even 𝑛. Therefore, 𝑇=π‘‡οŠοŠ°οŠ¨οŠ for all natural numbers 𝑛, so π‘˜=2.

We conclude that this sequence is periodic with order 2.

Let us finish by recapping some key concepts from this explainer.

Key Points

  • A recursive formula (sometimes called a recurrence relation) is a formula that defines each term of a sequence using a preceding term or terms.
  • A recursive formula of the form 𝑇=𝑓(𝑇) defines each term of a sequence as a function of the previous term. To generate a sequence from its recursive formula, we need to know the first term in the sequence, π‘‡οŠ§.
  • If we are given the first few terms of a recursive sequence, we can work backward from those terms to find a recursive formula for the sequence.
  • A sequence is increasing if 𝑇>π‘‡οŠοŠ°οŠ§οŠ for all natural numbers 𝑛.
  • A sequence is decreasing if 𝑇<π‘‡οŠοŠ°οŠ§οŠ for all natural numbers 𝑛.
  • A sequence is periodic if its terms repeat in a cycle. For any periodic sequence, there is an integer π‘˜ such that 𝑇=π‘‡οŠοŠ°ο‡οŠ for all natural numbers 𝑛. The integer π‘˜ is called the order (or period) of the sequence.
  • We can often tell whether a sequence is increasing, decreasing, or periodic simply by inspecting its recursive formula.

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